Derivatives in Sobolev spaces involving time

Question

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$, and consider for $T>0$ a function $u \in H^1(0,T;L^2(\Omega))$. Is it possible to speak about $u_t(T^-)$, i.e. the one-sided derivative in $T$ from the left? In general, is the expression $$ \int_\Omega u_t(T^-) u(T) \, dx $$ well defined?